Hyperbolicity is a group-theoretic property. That is, if AAA and BBB are finite sets of generators of a group GGG and Γ⁢(G,A)normal-ΓGA\Gamma(G,A) is a hyperbolic metric space, then Γ⁢(G,B)normal-ΓGB\Gamma(G,B) is a hyperbolic metric space.